boundary element formulation and its …etd.lib.metu.edu.tr/upload/12607232/index.pdfboundary...

BOUNDARY ELEMENT FORMULATION AND ITS SOLUTION IN

FORWARD PROBLEM OF ELECTROCARDIOGRAPHY BY USING A

REALISTIC TORSO MODEL

ARDA KURT

April 2006

BOUNDARY ELEMENT FORMULATION AND ITS SOLUTION IN

FORWARD PROBLEM OF ELECTROCARDIOGRAPHY BY USING A

REALISTIC TORSO MODEL

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF APPLIED MATHEMATICS

OF

THE MIDDLE EAST TECHNICAL UNIVERSITY

BY

ARDA KURT

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF

MASTER OF SCIENCE

IN

THE DEPARTMENT OF SCIENTIFIC COMPUTING

April 2006

Approval of the Graduate School of Applied Mathematics

Prof. Dr. Ersan Akyıldız

Director

I certify that this thesis satisfies all the requirements as a thesis for the degree

of Master of Science.

Prof. Dr. Bulent Karasozen

Head of Department

This is to certify that we have read this thesis and that in our opinion it is fully

adequate, in scope and quality, as a thesis for the degree of Master of Science.

Prof. Dr. Gerhard Wilhelm Weber

Supervisor

Examining Committee Members

Prof. Dr. Bulent Karasozen

Prof. Dr. Gerhard Wilhelm Weber

Prof. Dr. Nevzat Guneri Gencer

Assist. Prof. Dr. Yesim Serinagaoglu Dogrusoz

Assist. Prof. Dr. Hakan Oktem

I hereby declare that all information in this document has been ob-

tained and presented in accordance with academic rules and ethical

conduct. I also declare that, as required by these rules and conduct,

I have fully cited and referenced all material and results that are not

original to this work.

Name, Last name: Arda Kurt

Signature:

iii

Abstract

BOUNDARY ELEMENT FORMULATION AND ITS

SOLUTION IN FORWARD PROBLEM OF

ELECTROCARDIOGRAPHY BY USING A REALISTIC

TORSO MODEL

Arda Kurt

M.Sc., Department of Scientific Computing

Supervisor: Prof. Dr. Gerhard Wilhelm Weber

April 2006, 74 pages

The electrical currents generated in the heart propagate to the outward di-

rection of the body by means of conductive tissues and these currents yield a po-

tential distribution on the body surface. This potential distribution is recorded

and analyzed by a tool called electrocardiogram. It is not a problem, if this pro-

cess continues normally; however, when it is distorted by some abnormalities,

the results will be fatal. Electrocardiography (ECG) is the technique dealing

with the acquisition and interpretation of the electrical potentials recorded at

the body surface due to the electrical activity of the heart. This can be realized

by using one of the two approaches utilized in ECG namely; forward and inverse

problems. The former one entails the calculation potentials on the body surface

from known electrical activity of the heart and the latter one does the reverse.

In this thesis, we will construct the body surface potentials in a realistic torso

model starting from the epicardial potentials. In order to solve the forward

iv

problem, one needs a geometric model that includes the torso and the heart

surfaces, as well as the intermediate surfaces or the intervening volume, and

some assumptions about the electrical conductivity inside the enclosed volume.

A realistic torso model has a complex geometry and this complexity makes it

impossible to solve the forward problem analytically. In this study, Boundary

Element Method (BEM) will be applied to solve the forward problem numeri-

cally. Furthermore, the effect of torso inhomogeneities such as lungs, muscles

and skin to the body surface potentials will be analyzed numerically.

Keywords: Forward problem, boundary element method, realistic torso model.

v

Oz

GERCEKCI GOVDE MODELI KULLANARAK SINIRLI

ELEMAN YONTEMI ILE

ELEKTROKARDIYOGRAFIDE ILERI PROBLEM

COZUMU

Arda Kurt Yuksek Lisans, Bilimsel Hesaplama.

Tez Yoneticisi: Prof. Dr. Gerhard Wilhelm Weber

Nisan 2006, 74 sayfa

Elektrokardiyografide ileri problem arastırmasının temel nedeni, kalbin bili-

nen elektriksel aktivitesinden faydalanarak vucut yuzeyindeki elektriksel potan-

siyeli hesaplamaktır. Bu calısmada, giris cumlesinden de anlasıldıgı uzere,

kalbin elektriksel aktivitesinin gercekci bir govde modeli uzerindeki elektriksel

potansiyeli yapılandırılacaktır. Bu ileri problemi cozmek icin, hem kalbi hem

de govdeyi kapsayan geometrik bir modele ihtiyac oldugu gibi, bu iki yapının

arasında bulunan yuzeyler ve bu yuzeylerin elektriksel iletkenlikleri hakkında

bazı varsayımlara da gereksinim duyulmaktadır. Gercekci bir govde modelinin

karmasık bir geometriye sahip olması, ileri problemin analitik olarak cozumunu

guc kıldıgı icin Sınır Eleman Yontemi (BEM) kullanarak bu problemin sayısal

cozumu yapılacaktır. Ayrıca, vucudun homojen olmayan, akcigerler, kaslar ve

deri tabakası gibi yapılarının vucut yuzeyinde olusan elektriksel potansiyele etk-

ileri de tartısılacaktır.

Anahtar Kelimeler: Ileri problem, sınırlı eleman yontemi.

vi

To my family

vii

Acknowledgments

I would like to thank all those people who have helped in the preparation of

this study. I am grateful to Prof. Dr. Gerhard Wilhelm Weber (Willi) for his

motivation and support that provided me determination and power throughout

this study.

Special thanks to my co-advisor Assist. Prof. Dr. Yesim Serinagaoglu

Dogrusoz for her support and kindness.

I would also like to thank Prof. Dr. Bulent Karasozen, Prof. Dr. Nevzat

Guneri Gencer and Assist. Prof. Dr. Hakan Oktem for their valuable support

and time they spared for me.

And, I am grateful with my family so that I dedicated this study to them.

I am indebted to all of my friends, Lerzan, Mert, Emre, Ufuk, Tamer, Turan

for their patience and toleration to me during those hard days.

Thank you...

viii

Table of Contents

plagiarism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Oz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

ix

2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 The Heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Anatomy of the Heart . . . . . . . . . . . . . . . . . . . 3

2.1.2 Mechanical Activity of the Heart . . . . . . . . . . . . . 3

2.1.3 Electrophysiology of the Heart . . . . . . . . . . . . . . . 5

2.2 Forward and Inverse Problems of Electrocardiography . . . . . . 9

2.2.1 Methods in the Forward Problem of ECG . . . . . . . . 12

2.3 Forward Problem and BEM in the Literature . . . . . . . . . . . 14

2.3.1 Different Models . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Different Approaches . . . . . . . . . . . . . . . . . . . . 16

3 NUMERICAL SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Numerical Solution for the Homogeneous

Isotropic Torso Model . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Motivation of the Method . . . . . . . . . . . . . . . . . 17

3.1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.3 The Method of Solution . . . . . . . . . . . . . . . . . . 20

3.2 Numerical Solution for the Inhomogeneous Torso Model . . . . . 22

3.2.1 Motivation of the Method . . . . . . . . . . . . . . . . . 23

3.2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.3 The Method of Solution . . . . . . . . . . . . . . . . . . 24

4 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Method of Validation . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Validation of the Numerical Method . . . . . . . . . . . . . . . . 29

x

4.3 Numerical Solution for Realistic Models . . . . . . . . . . . . . . 35

4.4 Examining the Intermediate Matrices . . . . . . . . . . . . . . . 43

5 SUMMARY AND CONCLUSION . . . . . . . . . . . . . . . . . . . . 48

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

APPENDICES

A.1: Analytical Derivations for the Homogeneous Torso Model . . . . 58

A.2: Analytical Derivations for the Inhomogeneous Torso Model . . . 63

A.3: Calculation of the Solid Angle . . . . . . . . . . . . . . . . . . . 67

A.4: Integration over a Plane Triangle . . . . . . . . . . . . . . . . . 69

A.5: Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xi

List of Tables

4.1 The dipole locations and orientations used in the computations:

The values presented are in arbitrary units. . . . . . . . . . . . . 30

4.2 Spherical models used in the computations: The values are in

arbitrary units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Potential values of some points on the outermost surface calcu-

lated by analytical and numerical methods based on Model M21. 31

4.4 RMSE and RDME values for the potential values calculated by

analytical and numerical methods based on Model M22. . . . . . 32













4.11 Realistic models used in the computations: The values are in

arbitrary units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xii

4.12 The RDM* values for realistic models. . . . . . . . . . . . . . . 43

xiii

List of Figures

2.1 Location of the heart [8]. . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Anatomy of the heart [8]. . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Electrical activity of the heart [8]. . . . . . . . . . . . . . . . . . 6

2.4 A schematic of an electrophysiological recording of an action po-

tential showing the various phases which occur as the wave passes

a point on a cell membrane [56]. . . . . . . . . . . . . . . . . . . 7

2.5 Electrocardiogram signal [57]. . . . . . . . . . . . . . . . . . . . 8

2.6 Schematic representation of the forward and inverse problem. . . 10

2.7 Geometric description of the volume conductor. . . . . . . . . . 11

2.8 Eccentric spheres model [44]. . . . . . . . . . . . . . . . . . . . . 15

3.1 Homogeneous heart-torso model [7]. . . . . . . . . . . . . . . . . 19

3.2 Diagrammatic surface used for considering how to construct co-

efficient matrices P and G [7]. . . . . . . . . . . . . . . . . . . . 22

3.3 A volume conductor consisting of homogeneous isotropic lung

region in addition to heart and body surfaces [36]. . . . . . . . . 25

4.1 The graph of RDM* error evaluated for the radial dipole closest

to the innermost surface for several number of nodes. . . . . . . 36

4.2 The graph of computation time needed to compute the potential

distribution on the outermost surface for several number of nodes. 36

xiv

4.3 The inhomogeneous torso model used in the computations gen-

erated in map3d [27]. . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 Potential distribution on a randomly selected point on the body

surface for four different realistic models at several time instants. 39


surface for four different realistic models at several time instants. 40


surface for four different realistic models when the conductivity

of the muscles have been decreased from 0.675 to 0.600. . . . . . 41


surface for four different realistic models when the conductivity

of the lungshave been increased from 0.1 to 0.3. . . . . . . . . . 42

5.1 Solid angle subtended by a plane triangle at a point. . . . . . . 67

5.2 Illustration of the integral element dS/r. . . . . . . . . . . . . . 69

5.3 Adaptive numerical algorithm for evaluating surface integration

[23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xv

Chapter 1

INTRODUCTION

1.1 Motivation

The heart is the most considerable tissue of the body since it acts as a

battery for us to live; therefore it has to be well behaved and kept. There is no

problem when the heart is regularly working, however, in some cases when this

regularity is corrupted by arrhythmias that sometimes cause fatal heart diseases

(coronary heart disease, heart attack, congestive heart failure, endocarditis and

myocarditis, cardiac arrhythmia etc. [60]). Death rate extrapolations in the

USA for Cardiovascular diseases show that 945,836 people die every year, 78,819

per month, 18,189 per week, 2,591 per day, 107 per hour and 1 per minute [61],

and 1.5 million people die from cardiovascular diseases every year in the EU

[2]. It is projected that by 2025, approximately 60 % of the causes of death

worldwide will be due to cardiovascular and circulatory diseases [60, 61].

Good news is that the heart diseases today are much more treatable than it

used to be by the developing technology, progressive diagnosis and treatment,

and impressive new gains are on the near horizon. Roger Kamm, a mechani-

cal engineering professor who was himself involved in heart-disease studies in

Massachusets Institute of Technology (MIT), noted that there had been an ex-

plosion in the field of biomedical engineering over the past five years and the

evolving state of heart-disease research had led to a wide range of efforts in the

1

field at MIT [3]. Like Kamm, many other scientists have seen the heart research

so much worthwhile to deal for years. They have spent excessive effort and time

to find methods of preventing people from heart diseases in any field such as

bioinformatics, bioelectromagnetism, biomedical engineering, etc.

1.2 Purpose

Being a part of the area of heart research, a thesis work has been accom-

plished by me. This work includes the concepts of electrocardiography, forward

problem and boundary element method (BEM) using a realistic torso model in

order to deal with effects of the torso inhomogeneities to the computations.

1.3 Scope

This thesis report starts with background information including the me-

chanical and electrical activities in the heart, forward and inverse problems. It

continues with the forward problem formulations for a homogeneous heart-torso

model, and then, as an extension, with the forward problem calculations for an

inhomogeneous heart-torso model. The main objective of this report, on one

hand, is to present the effect of torso inhomogeneities to the potential distri-

bution on the body surface when compared with the inhomogeneous case. On

the other hand, the author shows how modern methods of applied mathematics

can be successfully employed in an important area of medicine and health care.

2

Chapter 2

BACKGROUND

2.1 The Heart

2.1.1 Anatomy of the Heart

The heart is a hollow, cone-shaped muscle located between the lungs and

behind the sternum. Two-thirds of the heart is located to the left of the midline

of the body and one-third is to the right (Figure 2.1).

The heart has 3 layers: The smooth inside lining of the heart is called the

endocardium. The middle layer of heart muscle is called the myocardium which

is surrounded by a fluid filled sac called as pericardium [8].

Inside of the heart is divided into four chambers, the two upper atria: Right

Atrium (RA), Left Atrium (LA) and the two lower ventricles: Right Ventricle

(RV), Left Ventricle (LV) (Figure 2.2).

2.1.2 Mechanical Activity of the Heart

The right atrium receives de-oxygenated blood from the body through the

superior vena cava (head and upper body) and inferior vena cava (legs and lower

torso). The tricuspid valve, which separates the right atrium from the right

ventricle, opens to allow the de-oxygenated blood collected in the right atrium

3

Figure 2.1: Location of the heart [8].

Figure 2.2: Anatomy of the heart [8].

4

to flow into the right ventricle. The right ventricle receives de-oxygenated blood

as the right atrium contracts. The pulmonary valve leading into the pulmonary

artery is closed, allowing the ventricle to fill with blood. Once the ventricles are

full, they contract. As the right ventricle contracts, the tricuspid valve closes

and the pulmonary valve opens. The closure of the tricuspid valve prevents

blood from backing into the right atrium and the opening of the pulmonary

valve allows the blood to flow into the pulmonary artery toward the lungs. The

left atrium receives oxygenated blood from the lungs through the pulmonary

vein. As the contraction triggered by the sinoatrial node progresses through

the atria, the blood passes through the mitral valve into the left ventricle.

The left ventricle receives oxygenated blood as the left atrium contracts. The

blood passes through the mitral valve into the right ventricle. The aortic valve

leading into the aorta is closed, allowing the ventricle to fill with blood. Once

the ventricles are full, they contract. As the left ventricle contracts, the mitral

valve closes and the aortic valve opens. The closure of the mitral valve prevents

blood from backing into the left atrium and the opening of the aortic valve

allows the blood to flow into the aorta and flow throughout the body [12].

2.1.3 Electrophysiology of the Heart

The mechanical activity described above is accompanied by electrical ac-

tivity in the heart. This activity begins with the Sinoatrial Node (SA Node)

(Figure 2.3), which is a group of cells positioned on the wall of the right atrium,

near the entrance of the superior vena cava. These cells have the ability to

generate electrical activity on their own [8, 59] which yields a change in the

electrical potential difference across the cell membrane. This dramatic change

in the electrical potential is known as an action potential (AP) [37]. SA Node

also known as the pacemaker. The heart also contains specialized fibers that

conduct the electrical impulse initiated in the SA node to the rest of the heart.

The electrical activity reaches the Atrioventricular Node (AV node), which is

the sole muscular connection between the atria and ventricles [38]. The electri-

5

Figure 2.3: Electrical activity of the heart [8].

cal impulse now goes to the Bundle of His and, then, it divides into the Right

and Left Bundle Branches where it rapidly spreads using Purkinje Fibers to the

muscles of the Right and Left Ventricle, causing them to contract at the same

time [8].

Action, Epicardial and Electrocardiogram Potentials

The source of the electric currents and potential distributions inside the

heart is the difference in the concentrations of positively and negatively charged

ions — especially, ionic potassium - across the cell membranes. Initially, the rest-

ing inside-to-outside potential, the transmembrane potential (TMP) difference,

is negative. A change of the ionic concentration could lead to a change in

the direction of the TMP. If more ions flow out, the consequence is a more

permeable membrane and a sudden breakdown of the resting potential, called

depolarization, and an action potential (AP) (Figure 2.4).

Once a cell is depolarized, it can trigger neighboring cells and cause them to

depolarize as well, producing a moving wavefront of varying electric potential,

which is called as activation wavefront. As a result of the ion movements during

6

Figure 2.4: A schematic of an electrophysiological recording of an action poten-tial showing the various phases which occur as the wave passes a point on a cellmembrane [56].

depolarization, the calcium concentration inside the cell increases, causing the

cell to mechanically contract, hence contracting the heart so that the blood is

squeezed upwards and outwards from the ventricles in an effective manner. After

one depolarization, there is an interval called the refractory period, during which

a second depolarization cannot occur until after some degree of repolarization

is complete. This ensures a minimum time interval between two consecutive

mechanical contractions.

The potentials observed on the surface of the heart as a result of the elec-

trical activities are known as epicardial potentials. These potentials differ from

the transmembrane potentials, but instead respond to extracellular potential

differences between regions of the myocardium.

Since the heart is in contact with the other compartments of the body, or

torso, such as lungs, blood vessels, muscles, etc., the electrical currents do not

stop, instead, propagate to the outward to the torso and their effect can be

measured as distributions of electrical potentials on the body surface which

are known as body surface potentials or electrocardiogram potentials. These

potentials are measured by a device called electrocardiograph by placing different

7

Figure 2.5: Electrocardiogram signal [57].

electrodes on the torso surface, which produces a graphical output showing

the electrical voltage in the heart in the form of a continuous strip, called

electrocardiogram [57]. Finally, electrocardiography (ECG, same abbreviation

for electrocardiogram) is the technique used to record the electrical impulses

which immediately precede the contractions of the heart muscle and relate the

epicardial potentials with the resultant body surface potentials.

An electrocardiogram produces waves that are known as the P , Q, R, S,

and T waves which gives each part of the ECG an alphabetical designation.

As the heart beat begins with an impulse from the SA node, the impulse will

first activate the upper chambers of the heart or atria and produce the P wave.

Then the electrical current will flow down to the lower chambers of the heart

or ventricles producing the Q, R and S waves. As the electrical current spreads

back over the ventricles in the opposite direction it will produce the T waves

(Figure 2.5).

Electrocardiogram is a common and relatively effective diagnostic tool and

in addition, it has the advantages of being cheap and painless [10]. By observing

the graph, the doctors

• can determine whether the heart is performing normally or suffering from

8

abnormalities (e.g., extra or skipped heartbeats - cardiac arrhythmia),

• may indicate acute or previous damage to heart muscle (heart attacks) orischemia of heart muscle (angina),

• can detect potassium, calcium, magnesium and other electrolyte distur-

bances,

• can detect conduction abnormalities (heart blocks and in bundle branchblocks),

• can provide information on the physical condition of the heart (e.g., leftventricular hypertrophy, mitral stenosis) [57].

2.2 Forward and Inverse Problems of Electro-

cardiography

As noted in the previous section, the main objective of electrocardiography

is to relate the potentials recorded at the body surface with the ones gener-

ated on the heart surface as a result of the electrical activity of the heart.

Two different approaches - in other words, problems — are used in electrocar-

diography to establish this relation. The first approach is called the forward

problem which entails the calculation of the body-surface potentials, starting

usually from either pre-decided electrical representation of the heart activity

(equivalent current dipoles) or from known potentials on the heart’s surface

(the epicardium) [20]. The other approach is the inverse problem, which in-

volves the calculation of the potentials and prediction of the electrical activity

of the heart starting from the potential distribution on the body surface, on

the contrary to forward problem (Figure 2.6). The general objective of the for-

ward and inverse problem of the electrocardiography is a better qualitative and

quantitative understanding of the heart’s electrical activity.

9

Figure 2.6: Schematic representation of the forward and inverse problem.

We have already stated that the current generated on the heart surface prop-

agates through other conducting compartments (lungs, sternum, spines, skeletal

muscle layer, and fat layer) of the torso so that the potential distribution mea-

sured on the body surface is directly affected by the geometrical and electrical

properties of the torso compartments. Each of these compartments has different

geometrical and electrical properties (conductivities). In the forward and in-

verse problem computations all these properties are taken into account in order

to have a precise solution at the end.

Although forward and inverse problems differ by definition, these two prob-

lems are closely related with each other, since the complexity of the physiological

torso model used must be tested through the forward solution and the inverse

solution must be developed from the forward solution. For a specific solution to

the forward or inverse solution of ECG, a three dimensional geometric descrip-

tion of a volume conductor (Figure 2.7) is required. The geometric description of

the volume conductor can take several forms, depending on the specific method

of mathematical solution [9].

When considering a realistic torso as in Figure 2.7, we have several regions

which are physically different and have different electrical conductivities. This

means that the volume conductor is inhomogeneous. Sometimes, in order to

simplify the calculations, the conducting medium between the heart and body

surface, inside of the torso, is assumed to be similar in geometry and have the

same conductivity values. In that case, the torso is described as homogeneous.

10

Figure 2.7: Geometric description of the volume conductor.

Moreover, if the conductivity value in a region changes with the direction then

the region is defined to be anisotropic, otherwise, if the conductivity is constant

over the region then the region is isotropic.

One of the key differences between the forward and inverse problems is

that the forward solutions are generally unique; on the other hand, the inverse

solutions are generally not unique [9]. Being not unique arises from the reason

that the primary cardiac sources cannot be uniquely determined as long as

the active cardiac region containing these sources is inaccessible for potential

measurements. This is because the electric field that these sources generate

outside any closed surface completely enclosing them may be duplicated by

equivalent single-layer (monopole) or double-layer (dipole) current sources on

the closed surface itself. Many equivalent sources and, hence, inverse solutions,

are thus possible. However, once an equivalent source (and associated volume

conductor) is selected, its parameters can usually be determined uniquely from

the body-surface potentials [20].

One drawback of the inverse problems is that, computationally, inverse prob-

lems frequently involve complex numerical algorithms and large systems of equa-

tions. In addition, inverse problems are also typically ill-posed, that is, small

changes in the input data can lead to deviations in the solutions. Often, the

major challenge of an inverse problem lies in incorporating a priori information

into the solution by regularizations and improvements [25]. The SCI Institute

11

has developed a number of efficient and realistic ways to solve a wide variety

of inverse problems in functional imaging - including reconstruction of elec-

trical sources within the heart or brain and extraction of molecular diffusion

information from magnetic resonance images [31].

The content of this study does not include the inverse problem, but the

forward problem of ECG. The above information has been given to make clear

the two problems of ECG and relations, however, one can find a discussion on

inverse problems in A.5.

2.2.1 Methods in the Forward Problem of ECG

The main motivation of forward problem of electrocardiography is to mea-

sure the body surface potentials from known epicardial potentials. This defini-

tion can be represented mathematically as follows:

PB = ZBH · PH . (2.2.1)

Here, PB is a vector of potentials on the body surface “to be measured”, PH

is a vector of “known” potentials on the heart surface, ZBH is a transformation

matrix determined from the geometrical coefficients and conductivities of the

torso regions and · denotes matrix multiplication.

Equation (2.2.1) is in the desired form for relating body surface potentials

to those in the heart since this equation expresses the vector of potentials on

the body surface as the product of a matrix of coefficients ZBH times the vector

of potentials on heart surface. In the remainder of this work, we will deal with

the methods/approaches used in the forward problem of ECG in order to build

the transformation matrix ZBH from the geometry and conductivities.

In the forward problem of electrocardiography, one of the two general ap-

proaches is used, namely, surface methods and volume methods.

In surface methods, the different torso regions are assumed to be of isotropic

12

conductivity, that is the conductivity is constant over the region, and only

the boundaries between the different regions are discretized and represented in

the numerical torso model. These computations entail the solution of integral

equations for the potential on the discretized surfaces of the torso model. Sur-

face methods are also termed boundary element methods (BEM), since only the

boundaries between torso regions are taken into consideration.

In volume methods, on the other hand, the entire three-dimensional torso

model is represented numerically, usually by a combination of the tetrahedral

and hexahedral (brick-shaped) elements. Volume methods may be subdivided

into finite-difference (FDM), finite-element (FEM), and finite-volume methods

(FVM).

Surface methods use simpler torso models with fewer elements. However,

since the underlying integral equations couple the potential at every element to

the potential at every element to the potential at every other element. Volume

methods use more complex torso models, with more elements and consequently

more potentials to be determined. However, the potential at each point is ex-

pressed only in terms of the potentials at its nearest neighbors. Consequently,

the coefficient matrix ZBH , while large, is also sparse. Volume methods repre-

sent the only way to incorporate individual regions of varying conductivity.

The FEM, which is a volume based method, requires a complete volume

mesh. The models that have used this method have tended to use a large number

of low order (linear) finite elements. This, for the three-dimensional case, results

in a large computational problem requiring vast computing resources. The FEM

does, however, have the advantage of being able to easily cope with anisotropic,

inhomogeneous regions.

The BEM, on the other hand, requires only a surface mesh. This reduces the

number of elements required and, hence, the size and complexity of the model.

The BEM also has the advantage in that it solves for the electrophysiologically

important gradients (or currents) directly. The main limitation of this method

for studying electrocardiography is that it can only be applied to isotropic re-

13

gions. However, there are works that applied BEM to anisotropic region as well

[62].

2.3 Forward Problem and BEM in the Litera-

ture

2.3.1 Different Models

Electrocardiographic potentials and effects of inhomogeneities of the differ-

ent torso regions have been studied theoretically, analytically and numerically

in the literature. Preliminary studies consist of only the theoretical and primi-

tive methods because of the lack of digital equipment and powerful computers.

Analytical studies utilized the simple geometrical shapes in order to model the

heart and the conducting volume where the heart is located. These studies

were important in giving an idea; however, they were not sufficient in show-

ing a real solution to the problem. Therefore, later studies used realistic heart

and torso models and used numerical methods by discretizing the regions taken

into account since analytical solutions cannot be applied to realistic geometries.

Although, recent studies brought extra work and computing effort, by the tech-

nology that has improved fast in last decades, this problem has been nearly

solved.

Most of the theoretical models of the heart-torso configuration were limited

to a consideration of only a portion of the internal inhomogeneities [44]. The

theoretical effect of the intracavitary blood mass on the body surface potentials

was first studied by Brody and its effect on the dipolarity of an equivalent

heart generator was investigated by Geselowitz and Ishiwatari. A theoretical

investigation of the effect of lung and blood inhomogeneities was performed by

Horacek, and by Arthur and Geselowitz. The surface muscle layer was modeled

as a flat planar layer by McFee and Rush.

In addition to the studies depicted above, Rudy and Plonsey [44, 45] studied

14

Figure 2.8: Eccentric spheres model [44].

the role of “all” internal geometry and inhomogeneities in the forward problem

by modeling the heart and the conducting volume where the heart is immersed

as “spheres” (Figure 2.8). An analytical solution, derived by considering only

the boundaries of the torso compartments, was found that demonstrates the im-

portance of interactions between the various torso compartments in determining

the potential distribution at the body surface.

In this model, the heart is represented as a sphere consisting of a central

blood volume bounded by a spherical heart-muscle cell and a pericardium; in

turn, the heart is located eccentrically within a spherical torso, where the latter

consists of a concentric lung region bounded by muscle and fat layers. Figure

2.8 shows the conductivity values (σ) for all regions.

The key idea under this approach is that the spherical geometry is clearly a

great idealization for obtaining an analytical solution to the problem. The main

advantage of an analytical solution (compared to numerical methods) is the

ability to include many layers in the model, and to manipulate the geometrical

parameters easily. A numerical method [7, 36] requires a new discretization

procedure whenever geometrical variations are introduced in the model.

15

Modeling as spherical or almost spherical geometry can be applicable to

estimate cortical potentials from scalp potentials [23] since geometry holds for

head model. However, in real world, heart and torso compartments do not have

concentric spherical geometry. For this reason, analytical computations derived

for these geometries do not hold for realistic heart and torso models. Anyway,

they have been widely used to check the accuracy of the numerical solutions in

the literature [33, 35, 46, 50].

2.3.2 Different Approaches

Gulrajani and Mailloux [19] studied inhomogeneity by successively adding

torso inhomogeneities to the torso model of Horacek which already incorporated

lungs and intraventricular blood masses which have conductivities relative to

the torso of 0.25 and 3.0, respectively. The first inhomogeneity added was the

anisotropic homogeneous skeletal muscle layer. The trick about the skeletal

muscle layer was that it had to be converted to an isotropic layer having a

conductivity value of 0.625 relative to torso by a series of computations in order

to be valid for boundary element calculations.

A better way to handle muscle layer and use the advantages of each of BEM

and FEM is to use a coupled FEM/BEM approach [39, 45] which has been

previously used by Stanley and Pilkington [51]. The model includes a number

of torso inhomogeneities, namely the epicardium, lungs, skeletal muscle layer

and subcutaneous fat layer. The BEM is used for isotropic and homogeneous

regions of the torso (e.g., the lungs and torso cavity), and the FEM is used for

anisotropic inhomogeneous regions (e.g., skeletal muscle layer).

16

Chapter 3

NUMERICAL SOLUTIONS

3.1 Numerical Solution for the Homogeneous

Isotropic Torso Model

The determination of the body surface potential distribution from known

epicardial potential distribution in a realistic-shaped torso is not applicable

to be solved analytically since the region in account (heart and body) does

not have a geometrically characteristic shape such as a sphere, ellipsoid, etc.

For this reason, one must use a systematic way of solving this forward problem

numerically by using one of the methods (FEM, BEM, FVM or FDM) described

before.

In this section, a numerical solution technique, which includes the discretiza-

tion of the regions in account and utilizes BEM, is presented. Based on this

numerical method, a Matlab program has been developed. In the final section,

the results of this computer program ran with a sample dataset will be discussed

and analyzed.

3.1.1 Motivation of the Method

As introduced before, the main goal of the forward problem of ECG is to

relate the body surface potentials with the epicardial potentials. Based on this

17

definition, we can redefine the forward problem mathematically analogous to

(2.2.1) as:

ΦB = ZBH · ΦH . (3.1.1)

Here, ΦB is a vector of potentials on the body surface “to be measured”, ΦH

is a vector of “known” potentials on the heart surface, ZBH is a transformation

matrix determined from the geometrical coefficients of the torso regions and ·denotes matrix multiplication.

Since, ΦH is already known and ΦB has to be determined, then one has to

derive a transformation matrix ZBH from the geometry of the torso regions in

order to establish a relation between body surface and epicardial potentials.

Based on a schematical heart-torso model presented in the next section,

Barr et al. [7] demonstrated the use of the boundary element method to find

the potential at the body surface from known epicardial potentials.

The method of Barr is based on knowing the geometric location of each

electrode, and on having enough electrodes to establish the geometric shape

and potential distribution of closed epicardial and body surfaces. The method

does not require that either the heart or body surfaces have any special shape,

such as a sphere, or that any electrical quantities, such as voltage gradients, be

known in addition to potentials.

The essence of this method is to find the coefficients that represent how

potentials on the outer surface of a volume conductor can be computed as a

linear combination of potentials on the inner surface. The methods takes into

account the consideration that the surfaces are defined by the known coordinates

of a set of points on each surface. It is assumed that the volume conductor

is homogeneous between the two surfaces, but that the conducting medium

terminates at the outer boundary.

18

Figure 3.1: Homogeneous heart-torso model [7].

3.1.2 The Model

The homogeneous heart-torso model (Figure 3.1) takes two regions into con-

sideration: The body surface (denoted as SB) and the heart surface (denoted

as SH) with unit normal vectors n.

The torso region of interest is the volume V surrounded by the heart surface

inside and the body surface outside. The inside volume is homogeneous and

isotropic with a conductivity value σ = σl. The outer part of the body surface

is air, so it has conductivity 0 (zero). Similarly, the inside region of the heart is

assumed to have a zero conductivity because all the potentials are accumulated

on the epicardium, the outer part of the heart. Here, the point O denotes an

observation point inside the volume and r denotes the distance between the

observation point and the surface element of integration inside the volume V .

Thiese definitions will be clearer in the following paragraphs.

Since, Figure 3.1 does not have a geometrical characteristic, such a sphere,

ellipsoid, etc., an analytical solution is not possible for this model. Hence, we

have to find a numerical solution for the transformation matrix ZBH by using

boundary element method as explained in the following subsection.

19

3.1.3 The Method of Solution

The method to be introduced in this section makes use of the BEM to solve

the forward problem numerically. Because of this reason, as the BEM implies,

the method only takes the “boundaries” of the heart and torso regions into

consideration.

The method starts with analytical derivation of integral equations by apply-

ing Green’s second identity to the volume V in order to reduce the number of

dimensions of the problem by one:

ZS

(A∇B −B∇A) · n · dS =ZV

¡A∇2B −B∇2A

¢· dV , (3.1.2)

where, as identified in (3.1), V is a volume surrounded by the surface S, and n

is an outward pointing vector of unit magnitude normal to the surface element

dS. In addition, A and B are two scalar functions of position. Surface S is

divided into two parts, one surrounded by the body surface, SB, and the other

surrounded by the heart surface, SH .

This integral equation can be applied to our model by substituting the two

scalar functions A and B in 3.1.2 with Φ, the scalar electric potential inside

the volume V , and 1/r, the reciprocal of the distance between the observation

point O and the surface of integration, respectively:ZS

µΦ∇

µ1

r

¶−µ1

r

¶∇Φ

¶·n·dS =

ZV

µΦ∇2

µ1

r

¶−µ1

r

¶∇2Φ

¶·dV . (3.1.3)

The details of the remaining analytical derivations, corresponding matrix

definitions and determination of the transformation matrix ZBH are presented

in A.1.

The final equation that relates the epicardial to body surface potentials is:

ΦB = ZBH · ΦH,

20

where

ZBH =¡PBB −GBHG

−1HHPHB

¢−1 ¡GBHG

−1HHPHH − PBH

¢. (3.1.4)

As mentioned before, the elements of matrix ZBH are the “transfer coefficient

matrices” [20] relating the potential at a particular epicardial surface point to

that at a particular body-surface point and depend purely on the geometry

of the epicardial and body surfaces. Here, the first subscript in each of the

matrices P and G denotes location of the observation point and the second

subscript denotes the surface where the integration is done.

Determination of Matrices P and G

The matrices P andG are coefficient matrices determined from the geometry.

A combination of these matrices forms the final transformation matrix relating

the epicardial potentials, ΦH , to the body surface potentials, ΦB.

The determination of matrices P and G is explained in [7] as the “iterative

close and distant regions” (Figure 3.2) approach. This approach starts with the

discretization in terms of triangles of the boundary surfaces and assumption that

the potential is constant over each triangle element. Then, the apexes of the

triangles are successively chosen as observation points. For every observation

point, its effect on the integration triangle on the integration boundary surface

is computed and the elements of the P and G matrices are evaluated as the

weighted sum of these effects.

Figure 3.2 shows the discretization procedure of the boundary surfaces. In

the first figure on the left, the surface is triangularized with the dot indicating

the observer location. In this figure, in the middle, the surface is divided into

“close” and “distant” regions. In other words, the total surface consisting of

triangles having the observation point as an apex point is the close region, SC ,

and each of the triangles that are not in the close region form the distant region,

SD.

21

distant SD

close

SC

k=1

k=NT

Figure 3.2: Diagrammatic surface used for considering how to construct coeffi-cient matrices P and G [7].

Having defined the close and distance regions, each coefficient matrix P and

G is determined in terms of the effects of each observation point on each of

the triangle element in the distant and close region. The detailed method for

determining each of the matrices P and G has been mentioned in [7].

One key point is that it is computationally advantageous to select as the

unknowns not the potential at each triangle but rather the potential at each

triangle vertex. Because, for most surfaces the number of triangle vertices is

approximately half the number of triangles [42, 43], thus diminishing the size

of the transformation matrix ZBH .

If the method for determining the coefficients of the P and G matrices in

[7] is observed well, the two important aspects can be seen from the integral

equations: The evaluation of solid angle (Ω) subtended by a triangle at an

observation point (see Appendix A.3) and integration over a plane triangle

dS/r (see Appendix A.4).

3.2 Numerical Solution for the Inhomogeneous

Torso Model

Having discussed the results of the forward problem using a homogeneous

heart-torso model, we can, now, extend our study to a more realistic case.

The model and the corresponding method introduced in the previous chapter

22

considered only two regions, heart and body surfaces. Since the homogeneous

model assumes that the region between the heart and body surface is of the same

property, the potential distribution on the body surface is not affected by the

conductivity of the internal torso, which is not the case in reality. We know that

the body consists of several compartments (lungs, bones, blood cavities, skeletal

muscle layer, etc.) which have different geometrical and electrical properties.

In realistic forward problem calculations, these regions have to be taken into

account.

In this chapter, a numerical method based on the previous method with

an extension of adding a homogeneous isotropic lung region to the previous

homogeneous model is presented. A Matlab program has been developed for

this inhomogeneous case and in the final section of this chapter; the results

obtained from this computer program will be discussed and analyzed.

3.2.1 Motivation of the Method

Recalling from previous section, the main goal of the forward problem of

ECG is to relate the body surface potentials with the epicardial potentials,

which is mathematically defined as:

ΦB = ZBH · ΦH

where ΦB is a vector of potentials on the body surface “to be measured”, ΦH

is a vector of “known” potentials on the heart surface, and ZBH is a transfor-

mation matrix determined from the geometrical coefficients and the electrical

conductivity values of the torso regions.

Since, ΦH is already known and ΦB has to be determined, one has to derive

a transformation matrix ZBH from the geometry of the torso regions in order

to relate the body surface and epicardial potentials. Differently from the ho-

mogeneous case, the transformation matrix, ZBH , is expected to depend on the

geometrical property of the lungs, in addition to heart and body regions.

23

Another difference arises from the conductivities of the regions. Unlike the

homogeneous heart-torso model, we have to take the conductivity values of the

heart, body and isotropic homogeneous lung region into consideration, because

the transformation matrix depends on the conductivities of the different regions.

Based on the definitions in the previous two paragraphs, the method of

Barr et al. [7] has been extended to a more general one that can handle the

inhomogeneities introduced in the model in the next section.

3.2.2 The Model

In order to relate the body-surface potentials to epicardial potentials in an

inhomogeneous volume conductor, the model previously introduced by Barr et

al. containing only the heart and body surfaces is now extended to have a

homogeneous isotropic compartment, lung (Figure 3.3).

The surfaces labeled SH , SL and SB correspond to heart, lung, and body

surfaces, respectively. The labels σH , σL, and σB correspond to the conductiv-

ity values in the same manner. Additionally, the signs “+” and “—“ show the

conductivity values in the exterior and interior parts of the region in consider-

ation.

Figure 3.3 can be further extended to have more inhomogeneities such as

blood cavities, subcutaneous fat layer, epicardium, etc.

3.2.3 The Method of Solution

In general, the problem to be solved for the inhomogeneous case is analogous

to the one of the homogeneous case. Motivation is to find a transformation

matrix formed by geometry and conductivity properties of the different torso

compartments which relates epicardial to body surface potentials.

The mathematical development of the transformation matrix for piecewise

homogeneous torso is similar to that of the homogeneous formulation of Barr

24

Figure 3.3: A volume conductor consisting of homogeneous isotropic lung regionin addition to heart and body surfaces [36].

et al. [7], although conductivity interfaces are considered. Detailed derivations

can be reached in Appendix A.2.

The relation between the body surface potentials with the epicardial poten-

tials can be summarized by the following equation [36]:

ΦB = ZBH · ΦH .

Here, ΦB and ΦH are vectors of torso and epicardial potentials, ZBH is a set

of transfer coefficients which depend only on the geometry and conductivities

of the epicardial and lung surfaces and · denotes matrix multiplication.

The resulting set of transfer coefficients relating body surface potentials to

25

epicardial potentials is expressed by:

ZBH = (¡PBB −GBHG

−1HHPHB

¢(3.2.5)

+¡PBL −GBHG

−1HHPHL

¢·¡PLL −GLHG

−1HHPHL

¢−1·¡GLHG

−1HHPHB − PLB

¢)−1

· ((GBHG−1HHPHH − PBH)

+ (GBHG−1HHPHL)

· (PLL −GLHG−1HHPHL)

−1

· (GLHG−1HHPHH − PLH)).

One can easily detect the analogy between the intermediate P and G matri-

ces that form the transformation matrix in the inhomogeneous case and the one

in the homogeneous case. The method of determination of P and G matrices is

similar to that of the homogeneous case.

Stanley et al. [36] demonstrated the effect of including more homogeneities

to the model introduced by Barr et al. [7]. The method of deriving the numerical

solution when more inhomogeneities are included is similar to the one given in

this section. The only difference is the increasing number of computations.

26

Chapter 4

RESULTS AND DISCUSSION

4.1 Method of Validation

In order to check the accuracy of the numerical solution derived in the pre-

vious chapters was compared with an analytical solution for concentric spheres

introduced in [47]. The analytical solution utilizes a dipole with strength and

orientation parameters and by using a series of weighted Legendre polynomials

to find the electrical potential on the surface of the outermost sphere. Since,

our numerical method relates the electrical potential on the outermost surface

with the innermost surface, a potential distribution on the innermost surface is

needed with the one on the outermost surface. Therefore, some changes were

made on the analytical solution so that it provides a distribution of electrical

potential on the given radius.

The procedure to check the accuracy of the numerical solution is as follows:

1. An electrical potential distribution has been computed on the innermost

(the heart) and the outermost surface by using the analytical solution due

to a dipole source model.

2. By using the numerical method, the transformation matrix ZBH has been

evaluated using only the geometry and conductivities.

27

3. The potential distribution on the heart surface, PH , has been multiplied

by ZBH to find the potential distribution on the body surface, PB.

4. PB found in step 3 has been compared with the one evaluated in Step 1

by using the analytical solution.

The above procedure has been repeated for different dipole strengths and

orientations, conductivity and radii values, and qualitative comparisons has

been made including inhomogeneities to the model.

In order to validate the numerical solution, first, for given radii and conduc-

tivity values, the concentric spheres has been generated and, then, the potentials

on the nodes on the innermost (matrix PH) and outermost surfaces (matrix PB)

have been computed. Next, the same nodes have been triangulated to be used

for the numerical solution. The triangulation has been done in such a way that

the points on the line passing through the origin and intersecting the surfaces

have been numbered the same. In other words, the point 81 on the outermost

surface and same numbered point on the innermost and other surfaces are lo-

cated on the same line that is passing through the origin. After the triangulation

procedure, the transformation matrix, ZBH , has been evaluated for the model.

It has been expected to get a similar result for PB, when multiplying the matrix

PH with the matrix ZBH .

The accuracy of the numerical solution has been tested by using Relative-

Difference-Measure-Error (RDME ):

RDME =

vuuuuut1n

nPi=1

(Pi − Ti)2

1n

nPi=1

T 2i

, (4.1.1)

where P is a vector of numerically computed potentials for all the nodes on the

body surface for a given dipole value, and T stands for the vector of analytically

computed potentials for the same nodes according to the same dipole value.

For a perfect fit, P = T and RDME = 0. RDME value shows how much

28

the “magnitudes” of the numerically computed potentials are different from the

magnitudes of the analytically computed potentials.

Another metric used for the accuracy check is Relative-Difference-Measure-

Error-Star (RDM* ):

RDMSE =

vuuuuutX⎛⎜⎜⎝ Pi

nPi=1

P 2i

− TinPi=1

T 2i

⎞⎟⎟⎠2

(4.1.2)

where Pij and Tij corresponds the ones in equation (4.1.1). Unlike RDME,

RDMSE gives an idea about how much the “potential distribution patterns”

received from the numerical differ from the ones from the analytical solution.

The difference in the distribution patterns is much more important than the

difference in the magnitudes of the potentials.

Next, in the realistic torso model, various inhomogeneities have been in-

cluded in the model, and the effects of including or excluding the inhomo-

geneities have been discussed.

Finally, the intermediate matrices received during the numerical solution

have been examined based on efficiency and singularity.

4.2 Validation of the Numerical Method

The validation procedure has been introduced in the previous section. In

this section, based on the validation method, the accuracy of the numerical

method will be checked for different models and dipole locations of different

locations, strengths and orientations.

The dipole locations and orientations used in the compuatations have been

generated by moving a dipole from the origin to the surface of the innermost

layer step by step. The dipoles have both tangential and radial orientations.

Additionally, the models used in the calculations are listed in Table 4.2.

29

Table 4.1: The dipole locations and orientations used in the computations: Thevalues presented are in arbitrary units.

Dipole No Location (x, y, z) Orientation (x, y, z)

1 10, 0, 0 0, 0, 104 40, 0, 0 0, 0, 107 70, 0, 0 0, 0, 109 90, 0, 0 0, 0, 1019 30, 0, 0 10, 0, 021 50, 0, 0 10, 0, 023 70, 0, 0 10, 0, 024 80, 0, 0 10, 0, 0

Table 4.2: Spherical models used in the computations: The values are in arbi-trary units.

Rad Cond NoN NoT

M21 100, 400 10, 1 132 260M22 100, 400 10, 1 380 756M23 100, 400 10, 1 501 998M31 100, 200, 400 10, 0.01, 1 132 260M32 100, 200, 400 10, 0.9, 1 132 260M33 100, 200, 400 10, 0.9, 1 501 998M41 100, 200, 300, 400 10, 0.9, 0.95, 1 132 260M42 100, 200, 300, 400 10, 0.9, 0.95, 1 380 756

In Table 4.1, the dipoles 1, 4, 7 and 9 are tangential; on the other hand,

the dipoles 19, 21, 23 and 24 are radial dipoles. In Table 4.2, M denotes

”the model”, the first subscript denotes the ”number of layers” (spheres) in the

model and the second denotes the ”index” of the model. Additionally, Rad,

Cond, NoN and NoT refer to the ”radii” of the layers from the innermost to

the outermost layer, ”conductivity” values of the layers in the same manner as

radii, ”number of nodes” and ”number of triangles” on each layer, respectively.

Table 4.3 shows the result of the computations with respect to M21 (Table

4.2). Since, it is not feasible to give the results for all the nodes and dipole values,

relatively more important cases have been chosen for discussion. In the following

tables, the header dipole refers to the dipole value used in the calculation, A and

30

Table 4.3: Potential values of some points on the outermost surface calculatedby analytical and numerical methods based on Model M21.

Dipole Analytic Numeric RDME RDMSE

1 -3.6425 -3.6932 1.4340 0.08004 -3.6093 -3.6380 0.8854 0.19477 -3.5387 -3.4886 1.2147 0.35449 0.0000 0.0015 1.1913 1.023721 -0.2011 -0.2323 1.7651 1.604924 -0.3311 -0.3654 1.8373 1.628326 -0.4551 -0.6659 7.8639 7.316327 -0.5143 -1.0807 19.9789 16.4658

N refers to the potential value at an arbitrary node computed by analytical and

numerical method, respectively. On the contrary, the RDME and RDMSE

values are not specific to a potential value computed for a specific point, but

for all the points on the outermost surface of the model.

Having observed the RDME and RDMSE values in Table 4.3, one can detect

that the sudden increase in the RDME and RDMSE values for the dipoles 23

and 24 which are radial dipoles closest to the surface of the innermost layer.

The reason for the inaccuracy yielded from the location of the dipole inside

the innermost surface is largely a quantization error. As the dipole location

approaches the inner shell surface the potential field on the inner shell surface

takes on an increasingly higher spatial frequencies. The node spacing on the

inner shell is inadequate to represent these narrow fields. The errror is greater

for a radial dipole than for a tangential dipole because the potential field is

more peaked [14]. Thus, the error should be reduced by increasing the number

of nodes representing the innermost sphere.

Table 4.4 shows the results for the same model but with more nodes and

triangles (model M22 in Table 4.2):

A more refined model, M23, has given more accurate results (Table 4.5):

As observed, the error has been decreased by the increasing the number of

nodes. The result is expectable, however, one should not forget that increas-

31

Table 4.4: RMSE and RDME values for the potential values calculated byanalytical and numerical methods based on Model M22.


1 -3.6425 -3.6509 0.2520 0.03084 -3.6093 -3.6144 0.1715 0.07927 -3.5387 -3.5418 0.1478 0.12219 0.0000 0.0002 0.3858 0.361119 -0.2011 -0.2071 0.4753 0.452221 -0.3311 -0.3343 0.4276 0.385923 -0.4551 -0.4475 0.5709 0.490324 -0.5143 -0.4517 3.1734 3.0119



1 -3.6425 -3.6480 0.1611 0.02314 -3.6093 -3.6125 0.1086 0.06077 -3.5387 -3.5403 0.0990 0.09359 0.0000 0.0001 0.2874 0.272219 -0.2011 -0.2055 0.3529 0.338621 -0.3311 -0.3334 0.3155 0.289523 -0.4551 -0.4501 0.4013 0.344824 -0.5143 -0.4931 1.2531 1.1405

32



1 -0.0866 -0.0852 1.5642 0.11304 -0.0859 -0.0841 2.1918 0.35387 -0.0845 -0.0810 4.2118 0.53529 0.0000 0.0016 4.2722 3.519219 -0.0041 -0.0432 78.4508 66.408921 -0.0067 -0.0517 89.8378 72.653423 -0.0092 -0.2054 385.7946 122.584024 -0.0104 -0.4616 879.3657 132.8023

ing the level of discretization means more computation time (10 seconds to 50

seconds in this example) [33].

Another discussion may arise from the relative conductivies of the compart-

ments inside the volume conductor. Table 4.6 presents the RDME and RDMSE

errors according to the model M31. The middle layer inserted into the model

M21 has a smaller conductivity value (0.01 units) than the one of the outermost

layer (1 units). When the conductivity of the middle layer is increased to 0.9

units (Model M32), the decrease in the errors is observable and the result is

corroborative to the idea that relatively small conductivity of one of the com-

partments compared with the conductivities of the other compartments yields

considerable numerical errors [33]. In Table 4.7, the errors have been consider-

ably reduced; however, further refinement can be done by increasing the number

of nodes in the model (Table 4.8).

The errors and refinements for four-layer models have been shown in Tables

4.9 and 4.10.

Based on the above observations and results, it can be concluded that the

accuracy of the numerical solution using BEM strictly depends on the dipole

locations and orientations, and the difference between the conductivities of the

layers. The former dependence arises when the dipole is a radial dipole (dipole

24) stationed very close to the surface of innermost layer. The tangential dipoles

33


Dipole Analytic Numeric RMDE RDMSE

1 -3.5016 -3.5435 1.2399 0.08234 -3.4700 -3.4910 0.6939 0.20407 -3.4027 -3.3486 1.4127 0.34939 0.0000 0.0015 1.1136 1.037619 -0.1917 -0.2228 1.7742 1.692621 -0.3156 -0.3495 1.8469 1.730723 -0.4338 -0.6430 8.2192 7.755824 -0.4903 -1.0504 20.7340 17.3577



1 3.5016 3.5050 0.1094 0.02394 3.4700 3.4712 0.0736 0.06427 3.4027 3.4026 0.1026 0.09969 0.0000 0.0001 0.2787 0.276019 -0.1917 -0.1960 0.3556 0.352721 -0.3156 -0.3178 0.3085 0.300223 -0.4338 -0.4286 0.3780 0.347124 -0.4903 -0.4683 1.2715 1.1844



1 3.5303 3.5720 1.2213 0.08254 3.4986 3.5192 0.6777 0.20517 3.4310 3.3761 1.4263 0.34829 0.0000 0.0015 1.1029 1.032819 -0.1929 -0.2238 1.7529 1.675821 -0.3176 -0.3512 1.8222 1.711023 -0.4366 -0.6456 8.1413 7.684024 -0.4935 -1.0539 20.5720 17.2147

34



1 3.5303 3.5336 0.1047 0.02404 3.4986 3.4997 0.0719 0.06477 3.4310 3.4308 0.1046 0.10049 0.0000 0.0001 0.2768 0.274719 -0.1929 -0.1972 0.3520 0.349721 -0.3176 -0.3197 0.3051 0.297823 -0.4366 -0.4312 0.3750 0.346024 -0.4935 -0.4714 1.2617 1.1761

(dipole 9) do not affect the solution as the radial dipoles, because, the radial

dipoles result a more peak potential. This dependence can be surpassed by

increasing the number of nodes, relatively triangles in the model. Beside the

increment of the number of nodes affect the accuracy in a positive manner, this

solution may result in excessive computation time for the numerical method.

Figure 4.1 represents how the refinement in the model affects the accuracy of

the solution. In addition, Figure 4.2 shows the effect of refinement to the com-

putation time of the numerical method. By observing the figures, one should

make a choice on the degree of refinement to get reasonable results in reason-

able duration. Moreover, the later dependence is especially the main problem

of electroencephalography and magnetocardiography (EEG/MEG), where the

conductivity of the skull is relatively small when compared to the brain. This

problem may be solved by the isolated problem approach used in many studies

[16, 17]. However, ECG studies does not face with this kind of problem.

4.3 Numerical Solution for Realistic Models

In this section, the numerical method derived in Chapter 3 has been uti-

lized to compute the body surface potential distribution for a “realistic torso

model”. The scheme presented in this section demonstrates the effects of torso

inhomogeneities to the body surface potential distribution by not physically

35

132160 196 236 289 380 501 702 8000

5

10

15

20

25

Number of Nodes

RD

M*

Err

or

Figure 4.1: The graph of RDM* error evaluated for the radial dipole closest tothe innermost surface for several number of nodes.

132160 196 236 289 380 501 702 8000

50

100

150

Number of Nodes

Com

puta

tion

time

(sec

)

Figure 4.2: The graph of computation time needed to compute the potentialdistribution on the outermost surface for several number of nodes.

36

Table 4.11: Realistic models used in the computations: The values are in arbi-trary units.

M1 M2 M3 M4

Comp. H, B H, L, B H, M, B H, L, M, BCond 10, 1 10, 0.1, 1 10, 0.675, 1 10, 0.1, 0.675, 1

including several inhomogeneities in the model. Instead, the models used in

the experiments includes only lung region in the volume conductor, but with

different conductivity values in each case. The models used in the computations

are mentioned in Table 4.11.

In Table 4.11,M denotes “the model” and the subscripts refer to the “model

number”. Additionally, Comp and Cond refer to the “compartments” included

in the model and the “conductivity” values of the compartments in order, re-

spectively. Finally, the labels H, L, M and B refers to the heart, lung, muscle

and body surfaces, in the same manner. The heart surface has 289 nodes and

574 triangles, the lung surface has 652 nodes and 1296 triangles, and the muscle

and body surfaces have 771 nodes and 1536 triangles.

An important note on the geometries of the compartments used in the follow-

ing computations is that only the torso was realistically shaped. Because of the

reason that the other compartments has been currently high-discretized (e.g.,

approximately 2000 nodes and 4000 triangles), the computation time would be

incredibly long, therefore, it was not feasible to use those models. Instead,

spherical geometries have been shifted, resized and rotated in or order to give

the corresponding form to the compartments as in Figure 4.3.

In order to analyze the effect of different torso inhomogeneities to the body

surface potential distribution on a realistic model, first, one needs an epicardial

potential distribution. In this study, the necessary epicardial distribution has

been generated by moving a dipole inside the heart. As a result, a potential

distribution for 230 time instants has been computed.

The numerical results received from those four models in Table 4.11 have

37

Figure 4.3: The inhomogeneous torso model used in the computations generatedin map3d [27].

been figured by using graphs as in Figures 4.4 and 4.5. The figures precisely

presents the effect of inhomogeneities to body surface potential distribution.

Moreover, four different graphs in the same figure provides a comparative anal-

ysis of the effect of inhomogeneities.

At first sight, one can state that the potential values received from both

models M1 and M2 are similar in magnitudes. Because, in both figures, the

value for the potential values computed for the same time instants are almost

the same or the difference is tiny. However, when the muscle region is in account,

M3, the change in the magnitudes of the potentials are apparent especially on

the magnitudes of the peak potentials which are increased by the addition of

the muscles. Finally, addition of both lungs and muscles, M4, has changed the

magnitudes of the potentials more than the single effect of muscles. Having in

mind that the relative conductivity of the lungs and muscles has been chosen

0.25 and 0.675, respectively, it has been expected that the effect addition of

the lungs should have been more visible than the effect addition of the muscles.

Because of the reason that the lungs do not surround the heart, while the muscle

region is parallel to the body surface and completely sorrounds the heart and

lungs, the change in the conductivity of the lungs do not considerably affect

the body surface potential distribution. However, the change in the potential

38

5 10 15 20 25 30 35 40 45−2

−1

0

1

2

3

4

Time

Pot

entia

lHeart + BodyHeart + Lung + BodyHeart + Muscle + BodyHeart + Lung + Muscle + Body

Figure 4.4: Potential distribution on a randomly selected point on the bodysurface for four different realistic models at several time instants.

magnitudes becomes considerable in case of a a small change in the conductivity

of the muscle region (Figure 4.6). Note that the peak potentials in Figure 4.6

have been increased when compared with Figures 4.4 and 4.5.

The same observation may be done by changing the effect of the change of

the conductivity of the lungs to body surface potential distribution. In Model

M2, the conductivity of the lungs was chosen as 0.1. Figure 4.7 shows the

resultant potentials for the same model, but the conductivity of the lungs has

been chosen as 0.3 which is three times of the original. One should note that

the peak potential have not been changed as much as in the case of decreasing

the conductivity of the muscles.

Another discussion may be arosen on the change of potential distribution

patterns. The body surface potential maps are not placed in this report, because

39

5 10 15 20 25 30 35 40 45−2

−1

0

1

2

3

4

Time

Pot

entia

l

Heart + BodyHeart + Lung + BodyHeart + Muscle + BodyHeart + Lung + Muscle + Body

Figure 4.5: Potential distribution on a randomly selected point on the bodysurface for four different realistic models at several time instants.

40

5 10 15 20 25 30 35 40 45−5

0

5

10

Time

Pot

entia

l


Figure 4.6: Potential distribution on a randomly selected point on the bodysurface for four different realistic models when the conductivity of the muscleshave been decreased from 0.675 to 0.600.

41

5 10 15 20 25 30 35 40 45−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time

Pot

entia

l


Figure 4.7: Potential distribution on a randomly selected point on the bodysurface for four different realistic models when the conductivity of the lungshavebeen increased from 0.1 to 0.3.

42

Table 4.12: The RDM* values for realistic models.

M1 M2- M1 M3- M1 M4 - M1

RDM* - 1.535 6.294 6.257

the black-white print-out makes the low-resolution figures useless. Instead, the

RDM* error analysis used in the previous section has been utilized to observe

the change in the potential distribution. Table 4.12 shows the numerical results

from RDM* analysis for all the points at all time instants. In other words,

the RDM* values have been calculated as the mean value of the RDM* values

for all the points for each of the 230 time instances. It can be concluded that

the addition of muscles have yielded a considerable change in the potential

distribution patterns, while the addition of lungs have a smaller effect than the

one of muscles.

As a result of the discussions in this section, the main conclusion is that

the effect of the muscles to the potential magnitudes and distribution patterns

on the body surface is more visible than the one of lungs. The possible reason

is that the muscle region completely surrounds the other compartments, while

the lungs are simply a seperated compartment like the others (i.e., heart, spine,

sternum) and their effect is less than the one of muscle region. Another possible

reason is that the space covered by the muscles is larger than the one of the

lungs in the torso.

4.4 Examining the Intermediate Matrices

Remembering from the Equation (3.1.4) and (3.2.5), ZBH is computed by

multiplication, addition and, the most important, the inversion of some inter-

mediate matrices. Each of these intermediate matrices is numerically computed

by using the geometry and conductivity properties. The evaluation of these

matrices includes the computation of solid angles subtended by triangles at ob-

servation points and the integrations over the area elements. These concepts

43

are discussed in Appendices A.3 and A.4.

In this section, the structure and evaluation of these matrices will be dis-

cussed.

The numerical solution was previously coded by using Matlab, which was a

big fault. Because of the reason that, each (i, j) element of these matrices is

computed by making a series of numerically discretized integral operations, the

whole matrix is built up in i×j iterations. Since Matlab is weak on element-by-

element computations, the first task was to transform the code into C++. By

this modification in the program, the computation time considerably decreased

from tens of minutes to a few seconds. To give an example, the computation

time to evaluate the transformation matrix for 3 spheres with 132 nodes and

260 triangles decreased from half an hour to 10 seconds. On the other hand, the

matrix manipulation operations have been carried in Matlab; because, Matlab

is extremely fast when compared to any program or tool. It is a common fact

that Matlab has implicit fast and efficient matrix manipulation and optimization

routines which makes it indispensable even for the cases when the matrices are

large. This is explicitly proved during the thesis study by comparing the speed

of matrix multiplication operations between two methods: one is Strassen’s ma-

trix multiplication coded in C++ and the other is simple matrix multiplication

in Matlab. The adjective “simple” is used literally meaning that multiplication

is simple in Matlab. Although, the Strassen’s matrix multiplication algorithm

is O(n2,81), which is more efficient than the classical matrix multiplication algo-

rithm of O(n3), the performance of Matlab was very reasonable. In fact Matlab

is capable of multiplying two 500×500 matrices in a reasonable time, while theother exhausts. As a result, after numerically computing the intermediate ma-

trices in element-by-element manner in C++, the final transformation matrix

is evaluated in Matlab.

The other criterion is the matrix inversion which is so sensitive to numerical

errors especially when the matrices are sparse and ill-conditioned. Although,

BEM is that BEM does not result in sparse matrices when being used in the

44

forward problem, the inversion problem has to be taken into consideration.

Therefore, instead of directly taking the inverse of a matrix, one should solve a

linear system of equations in closed form as follows:

A ·X = B, (4.4.3)

whereX can be replaced as the inverse of matrix A, A−1, and B as the identitity

matrix, I, for this work.

Successive elimination of variables, termed as Gaussian elimination, can

be used to solve (4.4.3). However, this operation is O³n4

3+ n3

2

´. Instead, a

decomposition scheme, which is the subject in consideration here, can be more

useful and efficient. This technique is called LU decomposition because of the

reason that it results in two triangular matrices such that:

A = L · U, (4.4.4)

where L is a lower triangular matrix and U is an upper triangular matrix.

This decomposition is useful, because the solution of triangular matrices

is easily accomplished by successive substitution in the corresponding linear

equations (starting with the corner of the triangle corresponding to a single

non-zero term on the left side of the equation) as follows:

L · U ·X = B, (4.4.5)

L · Y = B,

U ·B = Y.

First, the second sub-equation in (4.4.5) U ·X = Y is solved for Y by back

substitution and then the third sub-equation is solved for X by the same ap-

proach [55]. The decomposition and back substitution procedures take O³4n3

3

´time, which is more efficient than Gaussian elimination.

Another advantage of the LU decomposition is that the resulting triangu-

45

lar matrices L and U can be overwritten on the matrix A if the matrices are

computed such that the diagonal entries of the matrix L are 1. This situation

makes the LU decomposition a “space-efficient” method because the diagonal

entries of L is not need to be kept (4.4.6).

⎡⎢⎢⎢⎢⎢⎣A11 A12 A13 A14

A21 A22 A23 A24

A31 A32 A33 A34

A41 A42 A43 A44

⎤⎥⎥⎥⎥⎥⎦ =⎡⎢⎢⎢⎢⎢⎣

U11 U12 U13 U14

L21 U22 U23 U24

L31 L32 U33 U34

L41 L42 L43 U44

⎤⎥⎥⎥⎥⎥⎦ . (4.4.6)

The algorithm to find each Lij and Uij elements for a matrix A which has i

rows and j columns has been given in equation (4.4.7) [18] as follows:

Lij = U−1jj

µAij −

j−1Pk=1

LikUkj

¶j ≤ i− 1

Uij = Aij −i−1Pk=1

LikUkj j > 1. (4.4.7)

It can be observed from that the algorithm may run into numerical inac-

curacies if any Ujj becomes very small in the first line of (4.4.7). Thus the

absolute values of Uii are maximized if the rows of A are rearranged so that

the absolutely largest elements of A in each column lie on the diagonal. The

equations will become unchanged when the rows are rearranged, only the order

will change. This approach is called pivoting.

In this study, Matlab’s lu() function has been utilized to find the inverses of

the matrices such that:

[L,U, P ] = lu(A), (4.4.8)

where P ·A = L ·U . In (4.4.8), P is a permutation matrix to handle pivoting.

Then, the inversion scheme takes the following form:

L · U ·A−1 = P,

46

where

P = L · y,

U ·A−1 = y.

The first of the above equations can be solved for y and, later, the second

can be solved for A−1.

47

Chapter 5

SUMMARY AND

CONCLUSION

The heart is the most important tissue in the body, because it acts as a

battery. It pumps the clean oxygenated blood, which is needed by tissues for

respiration, to the body by the help of the surrounding epicardial muscles.

This operation is repeated periodically, approximately 80 times per minute in

a healthy person at rest. This mechanical activity in the heart is initiated and

driven by specialized cells which are capable of generating electrical currents.

The currents further move outward to the body surface, where they result a

potential distribution measured by medical tools. The potential distribution

generated on the body surface are called electrocardiogram potentials or body

surface potentials.

In order to understand the behaviour of the heart, its wellness or defections,

one needs an accurate relation between these potential distributions.The elec-

trocardiography is the technique to deal with the potential distributions on the

heart and on the body.

Electrocardiography utilizes two problems: the forward problem and the

inverse problem. The main motivation of the former one is to estimate the

potential distributions on the body surface from the corresponding potentials on

the heart surface. The later one, inverse problem, deals with the determination

48

of the epicardial potentials from the measured body surface potentials.

Mathematically, the forward problem is represented as follows:

PB = ZBH · PH ,

where PB and PH are the matrices including the information of the potential

distribution on the body surface and on the heart surface, respectively. On the

other hand, ZBH is a transformation matrix determined from the geometry and

conductivity properties.

The forward problem mainly presents two kind of methods, namely surface

and volume methods. In surface methods, also called boundary element methods

(BEM ), only the boundaries of the torso regions are taken into consideration,

whereas, the volume methods needs the whole volume to be discretized.

Based on the above summary, the content of this study includes the BEM

formulation and its solution using realistic torso models. Since the analytical

solution is not possible for a realistic models, first, the problem has been solved

by using concentric spherical models. This method is widely referred in the

literature to check the accuracy of the numerical methods. The approach pre-

sented to check the accuracy of the numerical method derived in Chapter 3

has started with the positioning of the tangential and radial dipoles into the

innermost sphere and receiving a potential distribution on both the heart and

body surface. Next, the transformation matrix has been evaluated with the

numerical method. Finally, the potential distribution on the heart surface is

transformed to the one on the body surface and the results of both numerical

and analytical method have been compared.

The comparisons have pointed out that the accuracy of the BEM strictly

depends on the dipole positions and orientations, and the differences between

the conductivities of the layers in the model. When the dipole is radial and

close to the inner boundary and the conductivity of one of the layers is rela-

tively small when compared to others, the numerical method yields considerable

49

errors. The first problem can be solved by increasing the number of nodes (and

also triangles) on the innermost boundary, and the later, by isolated problem

approach, which has not been mentioned in this study.

After the validation of the numerical method, in order to analyze the effects

of the inhomogeneities in a realistic torso model, a few observartions have been

accomplished with the relatively higher discretized boundaries compared to ones

in the homogeneous case in order to decrease the possible numerical error.

The effects of torso inhomogeneities has been studied in many ways [19, 36,

44, 45, 62]. The general idea is that the presence of lungs has a minimal effect

on the body surface potential distribution. As in the general sense, this work

presents that the lungs have minimal effect in the relation between epicardial

and body surface potentials, while the muscles have more considerable effects

on body surface potential distribution.

This work consists of considerable mathematical information, besides its

biomedical engineering side. The discussions on matrix manipulation opera-

tions, such as multiplication and inversion, and the establishments on efficiency

constraints on building the matrices, the observations on the computation of

solid angles and integration over surfaces and discretization procedures etc.

light the way for the people who work or willing to work on these concepts in

their disciplines.

50

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in EEG and MEG with applications to visual evoked responses, Cornelis

Johannes Stok, 1986.

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solver for the complete electrode model in EIT using algebraic multigrid,

IEEE Trans. Med. Imaging, Vol. 24(5), pp. 577-83, May 2005.

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[49] Srebro, R., A modified boundary element method for the estimation of

potential fields on the scalp, IEEE Trans. Biomed. Eng., Vol. 43(6), pp.

650-3, June 1996.

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of Bioelectromagnetism, Vol. 1, No. 1, 1999.

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technique for electrocardiographic calculations, IEEE Trans. Biomed. Eng.,

Vol. 36(4), pp. 456-61, April 1989.

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solution of the forward problem in magnetoencephalography, Phys. Med.

Biol., Vol. 48, pp. 523—41, 2003.

[53] Van Oosterom, A. and Strackee, J., The solid angle of a plane triangle,

IEEE Trans. Biomed. Eng., Vol. 30, pp. 125-26, 1983.

[54] Van Oosterom, A., Solidifying the solid angle, J. Electrocardiol., Vol. 35,

pp. 181-92, 2002.

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57

APPENDIX 1

Analytical Derivations for the Homogeneous Torso Model

This section will present the formation of the transformation matrix, ZBH , that

relates the epicardial potentials with the ones of body surface, starting from

series of interal derivations and discretization of integrals into matrix forms.

Remembering from Figure 3.1, we have a homogeneous torso model including

a volume V surrounded by heart surface, SH , and body surface, SB. Barr

et al. [7] applied Green’s second identity to the volume V and after making

corresponding substitutions, received:ZS

µΦ∇

µ1

r

¶−µ1

r

¶∇Φ

¶·n·dS =

ZV

µΦ∇2

µ1

r

¶−µ1

r

¶∇2Φ

¶·dV , (A1.1)

where Φ is the scalar electric potential inside the volume V , and 1/r is the

reciprocal of the distance between the observation point O and the surface of

integration, respectively.

In this homogeneous model, and also in reality, there are no current sources exist

inside the volume, since the only current generator is heart. Consequently, from

Laplace’s equation ∇2Φ = 0 inside the volume V . Hence, the above equationbecomes:Z

S

µΦ∇

µ1

r

¶−µ1

r

¶∇Φ

¶· n · dS =

ZV

Φ∇2µ1

r

¶· dV . (A1.2)

Additionally, if the observation point O inside the volume V is away from the

integration surface, then ∇2¡1r

¢= 0 for all r 6= 0.

However, If r = 0, then we have a singularity because∇2¡1r

¢is undefined. Here,

we use the method in [29, 38] to solve the singularity as follows:

If r = 0, then the volume integral becomes:

−ZV

Φ∇2µ1

r

¶· dV = Φ (pS)C, (A1.3)

58

and

C =

ZVs

∇2µ1

r

¶· dV ,

where ps is the point of singularity and Vs is volume just including the singu-

larity. By divergence theorem, we get:

C =

ZVs

∇2µ1

r

¶· dV =

ZVs

∇∇µ1

r

¶· dV =

ZSs

∇µ1

r

¶· dS. (A1.4)

Using spherical coordinates, we get:

limr→0

ZSs

∇µ1

r

¶· dS = lim

r→04πr2

µ− 1r2

¶· dS = −4π. (A1.5)

Combining the equations, we end up with the following integral equation:ZS

µΦ∇

µ1

r

¶−µ1

r

¶∇Φ

¶· n · dS = −4πΦo, (A1.6)

or in another form:

Φo =1

4π

ZS

Φ · r · nr2

· dS + 1

4π

ZS

∇Φ · nr

· dS, (A1.7)

where r is a unit vector in the direction of r and Φo is the potential at point O.

Since, surface S is composed of two surfaces SH and SB (S = SH ∪SB), (A1.7)can be rewritten as a sum of two integral equations, one for the heart surface

and the other is body surface:

Φo = −1

4π

ZSH

ΦH ·r · nHr2

· dSH −1

4π

ZSH

∇ΦH · nHr

· dSH (A1.8)

+1

4π

ZSB

ΦB ·r · nBr2

· dSB +1

4π

ZSB

∇ΦB · nBr

· dSB.

59

Note that the minus sign in the first two elements of the equation comes from

the reverse direction of the unit normal nH of the inner surface SH due to the

outer surface SB. Finally, utilizing the fact that ∇Φ = 0 on the outer surface,SB, [38], gives us the following equation:

Φo = −1

4π

ZSH

ΦH ·r · nHr2

· dSH −1

4π

ZSH

∇ΦH · nHr

· dSH (A1.9)

+1

4π

ZSB

ΦB ·r · nBr2

· dSB.

The above analytic equation giving the potential value at an observation point

O in terms of potentials on the heart surface and the ones on the body surface

is useful but not applicable, because, evaluating these integrals is not easy. The

objective here is to produce a numerical method of finding the potential at a

point in terms of heart and body surface potentials.

The approach introduced in [7] allows the observation point to be successively

stationed at different locations on either SH and SB. Suppose that there are

NB locations on SB and NH locations on SH . Positioning the observation point

on the body surface first yields:

ΦiB = −

1

4π

ZSH

ΦH · dΩiBH −

1

4π

ZSH

∇ΦH · nHr

· dSH +1

4π

ZSB

ΦB · dΩiBB, (A1.10a)

and then on the heart surface yields:

ΦiH = −

1

4π

ZSH

ΦH · dΩiHH −

1

4π

ZSH

∇ΦH · nHr

· dSH +1

4π

ZSB

ΦB · dΩiHB, (A1.10b)

where

dΩief =

ref · nfr2ef

· dSf .

Here, i is the index of the observation point on the corresponding surface. dΩief

is the solid angle subtended at an observation point of the ith location on the

surface e by an area element on surface f . The computation of the solid angle

60

is shown in Appendix A.3.

Changing the order of elements in equations (A1.10a) and (A1.10b) results in

the following form:

−ΦiB+

1

4π

ZSB

ΦB ·dΩiBB−

1

4π

ZSH

ΦH ·dΩiBH−

1

4π

ZSH

∇ΦH · nHr

·dSH = 0, (A1.11a)

and

− 14π

ZSH

ΦH ·dΩiHH−Φi

H−1

4π

ZSH

∇ΦH · nHr

·dSH+1

4π

ZSB

ΦB ·dΩiHB = 0. (A1.11b)

Each of the terms of (A1.11a) and (A1.11b) can be discretized as follows:

−ΦiB +

1

4π

ZSB

ΦB · dΩiBB =

NBXj=1

pijBBΦjB, (A1.12a)

− 14π

ZSH

ΦH · dΩiBH =

NHXj=1

pijBHΦjH , (A1.12b)

− 14π

ZSH

∇ΦH · nr

· dSH =NHXj=1

gijBHΓjH , (A1.12c)

1

4π

ZSB

ΦB · dΩiHB =

NBXj=1

pijHBΦjB, (A1.12d)

−ΦiH −

1

4π

ZSH

ΦH · dΩiHH =

NHXj=1

pijHHΦjH , (A1.12e)

− 14π

ZSH

∇ΦH · nr

· dSH =NHXj=1

gijHHΓjH . (A1.12f)

Here, ΦH and ΦB are vectors containing the potentials at the NB and NH

locations on surfaces SB and SH , respectively, and ΓH contained the normal

components of the gradients of SH . The first superscript and first subscript for

each of the p’s and g’s identifies the location and surface where the observer

is stationed. The second superscript and subscript identifies the element of

61

corresponding surface of integration. The p’s and g’s are coefficients whose

values are determined on the basis of geometry.

Substituting (A1.12a) - (A1.12f) into (A1.11a) and (A1.11b) yields:

piBBΦB + piBHΦH + giBHΓH = 0, (A1.13)

piHBΦB + piHHΦH + giHHΓH = 0.

Here, i denotes the ith location, first, on surface SB and then on SH . (A1.13),

then, yields (A1.14) by chhosing location i successively at all NB locations on

SB, and on NH locations on SH :

PBBΦB + PBHΦH +GBHΓH = 0, (A1.14)

PHBΦB + PHHΦH +GHHΓH = 0.

Here, all P and G are coefficient matrices, which are determined from the

geometry (see Appendix A.1). These matrices are the “building stones” of the

final transformation matrix ZBH , that relates the potential distribution on the

heart surface to the ones on the body surface.

If we solve (A1.14) for ΦB in terms of ΦH , we receive (A1.15):

ΦB = ZBH · ΦH , (A1.15)

where

ZBH =¡PBB −GBHG

−1HHPHB

¢−1 ¡GBHG

−1HHPHH − PBH

¢.

Equation (A1.15) presents that ZBH is determined from a combination of coef-

ficient matrices which depend only on the geometry.

62

APPENDIX 2

Analytical Derivations for the Inhomogeneous TorsoModel

In Appendix A.1, a numerical method of finding a transformation matrix from

the geometry of a homogeneous volume conductor taking only heart surface

and body surface into consideration and assuming that the conducting medium

between these surfaces to be isotropic, has ben presented. Based on the as-

sumptions and derivations for a homogeneos volume conductor, in this section,

we will introduce a numerical method of finding a transformation matrix that

relates epicardial to body surface potentials for an inhomogeneous volume con-

ductor in terms of geometry coefficients and conductivity values.

The model has been extended from a homogeneous volume conductor to one

having an isotropic homogeneous lung region. Although, we have two lungs

as shown Figure 3.3, these two regions can be treated as one combined region

mathematically.

The method of finding a transformation matrix for an inhomogeneous volume

conductor is similar to one for the homogeneous case, although, we consider

conductivities.

Stanley et al. [36] made use of the Green’s second identity for the volume

V surrounded by the closed surface S, which is composed by surfaces of five

torso regions (heart, lungs, spine, sternum, body and skeletal muscle layer) and

substitute A with a scalar function 1/r and B with another scalar function not

Φ, but σΦ, where σ denotes the conductivity of the corresponding region.

If we replace Φ in (A1.6) of Appendix A.1 and remember that∇2(σΦ) = 0, since,there is no current source in the volume V , we have the following simplification:ZS

µ1

r∇ (σΦ)− σΦ∇

µ1

r

¶¶· n · dS = −

ZV

σΦ∇2µ1

r

¶· dV = 4πσoΦo, (A2.1)

where Φo denotes the potential value at an observation point in the volume V .

63

Since, surface S is consisting of three surface SH , SL and SB (S = SH∪SL∪SB),(A2.1) can be expanded into a series of integrals considering each surface as the

following:

−ZSH

µ1

rOHσB∇ΦH − (σB − σH)ΦH∇

µ1

rOH

¶¶· nH · dSH (A2.2)

−ZSL

µ1

rOL(σB − σL)∇ΦL − (σB − σL)ΦL∇

µ1

rOL

¶¶· nL · dSL

+

ZSB

µ1

rOBσB∇ΦB − (σair − σB)ΦB∇

µ1

rOB

¶¶· nB · dSB

= 4πσoΦo,

where ΦH , ΦL and ΦB denotes the potentials on heart, lung and body surfaces,

respectively. Similarly, σH , σL adn σB denotes the conductivity values of heart,

lung and body surfaces, respectively. Furthermore, rOH , rOL and rOB denotes

the distance r from an observation point O to a surface element on the heart,

lung and body surfaces, respectively. Finally, nH , nL and nB denotes the normal

vectors of the surfaces SH , SL and SB.

As a result of the reason that the body is surrounded by air, which is non-

conducting (σair = 0), the normal component of the current across the body

surface is zero:

σB∇ΦB · nB · dSB = 0.

Additionally, at the lung-body boundary, we have the following boundary con-

dition, since, the normal component of the current is continuous:

σL∇ΦL · nL = σB∇ΦL · nL.

We already know that ∇1r= −r·n

r2from Appendix A.2.

64

Based on the above considerations, we can simplify (A2.2) as follows:

Φo = −1

4π

σBσo

ZSH

1

rOH∇ΦH · nH · dSH (A2.3)

− 1

4π

σBσo

ZSH

ΦHdΩOH · dSH

− 1

4π

(σB − σL)

σo

ZSL

ΦLdΩOL · dSL

+1

4π

σBσo

ZSB

ΦBdΩOB · dSB,

where

dΩef =ref · nfr2ef

.

Here, Φo is the electrical potential value at an observation point in the volume

V , in terms of the potentials on the heart, lung and body. The notation dΩef is

the solid angle subtended by a surface element f at an observation point on the

surface e. The computation of the solid angle is represented in the Appendix

A.4.

Similar to the approach in the homogeneous case, suppose that there are NH

locations on surface SH , NL locations on surface SL and NB locations on SB

and let the observation point to be succesively stationed on NH , NL and NB

locations. After a series of substitutions and discretizations [36], we will receive

the following form:

PBBΦB + PBLΦL + PBHΦH +GBHΓH = 0, (A2.4)

PHBΦB + PHLΦL + PHHΦH +GHHΓH = 0,

PLBΦB + PLLΦL + PLHΦH +GLHΓH = 0.

Here, all P and G are coefficient matrices, which are determined from the

geometry and conductivities (see Appendix A.2) as in the homogeneous case.

If we solve equation (A2.4) for ΦB in terms of ΦH , we receive equation (A2.5):

65

ΦB = ZBH · ΦH , (A2.5)

where

ZBH = (¡PBB −GBHG

−1HHPHB

¢(A2.6)

+¡PBL −GBHG

−1HHPHL

¢·¡PLL −GLHG

−1HHPHL

¢−1·¡GLHG

−1HHPHB − PLB

¢)−1

· ((GBHG−1HHPHH − PBH)

+ (GBHG−1HHPHL)

· (PLL −GLHG−1HHPHL)

−1

· (GLHG−1HHPHH − PLH)).

Equation (A2.6) presents that ZBH is determined from a combination of coef-

ficient matrices which depend on the geometry and the conductivities.

66

APPENDIX 3

Calculation of the Solid Angle

The solid angle subtended by a surface at a point is defined as the surface area of

the projection of that surface onto a unit sphere centered at that point (Figure

5.1) and usually denoted by Ω. The P matrices defined in previous chapters

need the efficient computation of the solid angles.

Figure 5.1: Solid angle subtended by a plane triangle at a point.

In Figure 5.1, R1, R2 and R3 are three vectors from an observation point O to

the three vertices of a triangle element on the surface of integration, A is the

area of the triangle formed by the end-points of the vectors R1, R2 and R3 and

Ω is the solid angle subtended by surface area A at observation point O.

The calculation of the solid angle is one of the main loads in forming trans-

formation matrix [54]. An efficient algorithm for calculating the solid angle Ω

subtended by a triangle with vertices R1, R2 and R3, as seen from the origin

has been given by Oosterom and Strackee [53] in the following equation:

Ω = 2arctan

µ[R1R2R3]

R1R2R3 + (R1 ·R2)R3 + (R1 ·R3)R2 + (R2 ·R3)R1

¶.

(5.0.1)

67

Here, [R1R2R3] denotes the determinant of the matrix that results when writing

the vectors together in a row, Ri denotes the distance of point i from the origin,

Ri is the vector representation of point i, and Ri ·Rj denotes the scalar vector

product.

Equation (5.0.1), as the definition implies, is valid when the observation point

is at the origin. Because of this, before computing the solid angle, first, the

observation point has to be shifted to the origin by subtracting its coordinates

(xo, yo, zo) from the coordinates of each of the vertices of the triangle.

As well as being efficient, this algorithm has limitations on evaluating the solid

angle subtended by the triangle at an observation point which is also one of the

vertices of the triangle. This condition is named as auto solid angle, in which,

the denominator in (5.0.1) will be zero and the result of the division operation

will be undefined.

The accuracy of the BEM strictly depends on the calculation of solid angles,

especially auto solid angles. However, calculation of auto solid angle is singular,

as described above, thus, efficient methods [24, 33, 41] are needed.

In the numerical solution given in this study a modified version of the equation

(5.0.1) is used with [13] to improve the accuracy of the solution. The solid angles

are computed with linear discretization method and the effect of the solid angle

is distributed relatively between the nodes of the triangle in consideration, unlike

the simple method of Barr [7] in which the effect of the total solid angle was

distributed equally to the vertices of the triangle. The auto solid angles, on

the other hand, cannot be directly computed with the above equation because

of the singularity. Instead, they are computed by subtracting the solid angles

computed for the off-diagonal entries of the matrix P from the total.

68

APPENDIX 4

Integration over a Plane Triangle

The methods presented in this study to form the transformation matrices that

relate the epicardial to body surface potentials, ZBH , consists of evaluating the

surface integral dS/r from an observation point O distanced r units to a triangle

element dS (Figure 5.2).

Figure 5.2: Illustration of the integral element dS/r.

In addition to the calculation of the solid angle, the evaluation of the surface

integral dS/r is one of the main loads in forming transformation matrix. In the

simple cases when the observation point O is not on the integration surface S or

the resultant scalar value, the electrical potential, is assumed to be constant over

the integration surface, the integration is relatively simple and may even have

a closed analytical solution [7]. The potential value evaluated for the triangle

element is then assigned to the vertices of the triangle in some manner, for

example, equally in Barr et al. [7]. However, in such cases when the observation

point is on the integration surface, these integrals become singular — “worst”

integrals — , hence, modified techniques are needed to evaluate these integral

accurately. Because of the reason that these “worst” integrals are in general the

largest in magnitude and lead to the diagonally dominant matrix equation and

69

Figure 5.3: Adaptive numerical algorithm for evaluating surface integration [23].

have most influence on the solution, it is important to calculate these integrals

as accurately as possible [26] by using modified integration techniques.

A modified approach in which each original planar element is subdivided and

a constant value is assigned to each of the resulting subelements has been in-

troduced by [23]. The specifics of this technique depend on exactly how the

triangle is subdivided (Figure 5.3), and how potentials are assigned to the re-

sulting subelements. One important advantage of this method is that it permits

adaptive solutions in which subdivision occurs only where and when required,

resulting in increased computational efficiency.

A good approximation for the previous approach has been given by Oosten-

dorp and van Oosterom [35]. Instead of using the computation time consuming

analytical expressions for the calculation of the integral 1/r, the mean of the

values of 1/r at the vertexes of the triangle is multiplied by the area of the

subtriangle. Similar to the auto solid angle case mentioned in Appendix A.4, a

singularity arises here when r is 0, in other words, when the observation point

is one of the vertices of the triangle. By an approximation, this singularity is

solved by taking the value of 1/r at the center of mass of the subtriangle instead

of infinite value at the observation point. The same approach was utilized in

70

[49], where, the area integral was estimated as the area of each triangle divided

by the distance from the observation point on the scalp to the centroid of each

triangle. The potential was assumed constant over each triangle and equal to

the average potential over the three vertices (nodes) of the triangle.

A third approach to approximating the integrals in boundary elements is to al-

low the scalar value to vary over the surface element and then employ numerical

surface integration (numerical quadrature) techniques based on Radon [40] for

each element [11, 30, 36]. The numerical quadrature then becomes a weighted

average of the potentials at the vertices.

De Munck [13] presented more general choices to discretize the boundary inte-

gral equations, without destroying the analytical computation of the matrix el-

ements. The later study [46] implemented the approach presented by de Munck

in calculating the electric potential and investigating the effect of different dis-

cretization choices.

The surface integrations evaluated in this study are based on De Munck’s ap-

proach with modifications by Ferguson et al. [15].

In BEM, as well as the computation of the surface integral is important, one

should take attention on the discretization of the surfaces. If there is a large

number of points and triangles (i.e., hundreds, thousands) in the model, then

the numerical errors in computing surface integrals and solid angles may be

decreased. However, this increase in the number of discretization points will

result in extensive computation time and storage demands [33]. As a result,

in order to receive reasonable results from BEM, special care is needed for

computing surface integrals and solid angles.

71

APPENDIX 5

Inverse Problems

Aster et al. [4] defined a system as a composition of physical parameters and

observations. Scientists and engineers frequently need to relate the physical

parameters to observations of the system. Mathematically, a system can be

represented as:

G(m) = d, (A5.1)

where m refers to physical parameters or model, d refers to observations or data

and G is a function or forward operator that relates the model m to data d.

In applied mathematics, G(m) = d usually refers to the “mathematical model”,

and m refers to the “parameters” of the mathematical model.

The data d may be a continuous function of time and space, or may be a col-

lection of discrete observations. In addition, the model m may be a continuous

function of time and/or space, or it may be a collection of discrete parameters

as well. The operator G can take on many forms. In some cases, G is an or-

dinary differential equation (ODE) or partial differential equation (PDE). In

other cases, G might be a linear or nonlinear system of algebraic equations.

The data has two important issues. One of them is that the actual observations

always contain some amount of noise which may arise due to missing specifica-

tions in the model or numerical errors. The second issue is that there are also

commonly an infinite number of models that satisfy the data d, for the same G.

The forward problem is to find d given m by computing G(m), which might

involve solving an ODE, solving a PDE, evaluating an integral, or applying an

iterative or adaptive algorithm. The inverse problem, on the other hand, is to

obtain an estimate of m given d.

In the cases, where m and d are characterized as continuous functions of time

and space, associated task of estimating m from d is called a continuous inverse

problem.

However, in many cases, we will want to determine a finite number, of param-

72

eters, n, that define a model m. Then, we can express m as a vector of n

parameters. Similarly, the data d can be expressed an m-dimensioned vector

d. Such problems are called discrete inverse problems or parameter estimation

problems. A general discrete inverse problem can be written as a (possibly

nonlinear) system of equations:

G(m) = d, (A5.2a)

or

G ·m = d. (A5.2b)

Solving a forward problem is easy and straightforward; since, for a specific

model m and a forward operator G, computing the data d is obvious. Unlike

the forward problem solving an inverse problem, in other words finding mathe-

matically acceptable answers to inverse problems, is hard. Because, there may

be many models that adequately fit the data. In order to decide whether the

solution is acceptable or not, some important issues that must be considered

include existence, uniqueness, and instability of the solution.

1. Existence: There may be no model m that exactly fits a given data d.

2. Uniqueness: If exact solutions do exist, they may not be unique, even for

an infinite number of exact data d.

For example, let m1 and m2 be two solutions of the inverse problem that

m1 6= m2, then the following two equation can be acceptable due to the

same data d:

G ·m1 = d,

and

G ·m2 = d.

3. Instability: The process of computing an inverse solution is often ex-

tremely unstable. That is, a small change in measurement can result an

73

enormous change in the inferred model. Systems where small changes

in the data can drive large changes in inferred models are referred to

as ill—posed (in the case of continuous systems), or discrete ill—posed or

ill—conditioned (in the case of discrete linear systems).

Ill-conditioning of the problem arises from the situation that the operator G

(it is a matrix in discrete inverse problem) is sparse and has a large condition

number, singular values of G decay gradually to zero or the ratio between the

largest and smallest nonzero singular values is large.

An interesting and important aspect of ill-posedness is that the ill-conditioning

of the problem does not mean that a meaningful approximate solution cannot

be computed, however, means that the standard methods such as LU decom-

position, Cholesky or QR factorization cannot be used in a straightforward

manner. Instead of these numerical methods, more sophisticated ones - regu-

larization methods (Tikhonov regularization, quadratic least squares, singular

value decomposition) - must be applied to ensure the computation of a mean-

ingful solution [22]. Additionally, some preconditioning methods [34, 48, 52, 1]

should be used to make the problem well-conditioned by decreasing the condi-

tion number of the sparse matrix G, because computing with a sparse matrix

needs so much time and memory space.

74

boundary element formulation and its …etd.lib.metu.edu.tr/upload/12607232/index.pdfboundary...

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